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Section: Research Program

FT and FxT control and estimation

To design an estimation or control algorithm we have to select a performance criterion to be optimized. Stability is one of the main performance indexes, which has to be established during analysis or design of a dynamical system. Stability is usually investigated with respect to an invariant mode (e.g., an equilibrium, desired trajectory or a limit cycle), then another important characteristics is the time of convergence of the system trajectories to this mode, which can be asymptotic (in conventional approaches) or finite-time (being the focus of Non-A POST team). In the latter case the limit mode has to be exactly established in a finite time dependent on initial deviations (if such a time is independent on initial conditions, then this type of convergence is called fixed-time). If the rate of convergence is just faster than any exponential of time, then such a convergence is called hyperexponential. The notion of finite-time stability has been proposed in 60s by E. Roxin and it has been developed in many works later, where a particular attention is paid to the time of convergence for trajectories to a steady state (it is worth to note that there exists another notion having the same name, i.e. finite-time or short-time stability, which is focused on analysis of a dynamical system behavior on bounded intervals of time, and it is completely different and not considered here). For example, the following simple scalar dynamics:

has the solution $x\left(t\right)=\beta \left(\right|x_0|,t)\text{sign}\left(x_0\right)$ for any initial condition $x\left(0\right)=x_0\in ℝ$, where

which possesses a finite-time convergence with the settling time $-\frac{|x_0{|}^{-\nu }}{\nu }$ for $-1\le \nu <0$, an exponential convergence for $\nu =0$ and the fixed time of convergence to the unit ball is bounded by ${\nu }^{-1}$ for $\nu >0$. It is straightforward to check that this simple system is homogeneous of degree $\nu$. The members of Non-A POST team obtained many results on analysis and design of control and estimation algorithms in this context. A useful and simple method to deal with these three types of convergence (finite-time, fixed-time or hyperexponential) is based on the theory of homogeneous systems.